Category: Uncategorized

Horospherically invariant measures on geometrically infinite quotients

Abstract: We consider a locally finite (Radon) measure on invariant under a horospherical subgroup of where is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not admit any additional invariance properties then it must be supported on a set of points with geometrically degenerate trajectories under the corresponding contracting 1-parameter diagonalizable flow (geodesic flow). We deduce measure classification results. One such result in the context of finitely generated Kleinian groups in will be highlighted. Most of the talk will be based on joint work with Elon Lindenstrauss.

Random walks by homeomorphisms on the line and left-orderability of lattices

Abstract: The standard random walk in the integers is known to be recurrent, it passes through every integer infinitely many times. We will discuss a generalization of this theorem for random walks given by random composition of homeomorphisms of the line (due to Deroin-Navas-Kleptsyn-Parwani) and some applications of this theorem to the theory of left–orderable groups. Our main result is that a lattice in a real semi-simple Lie group of higher rank and finite center is not a left–orderable group (equivalently, every action of such lattice in the line by homeomorphisms is trivial), a conjecture due to Witte-Morris and Ghys.  (Joint work with Bertrand Deroin).

Slides